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Let \(M\) and \(N\) be differentiable manifolds and \(\varphi : M \longrightarrow N\)
a differentiable map. A mixed differential form along\(\varphi\) is an element
of the graded algebra represented by
MixedFormAlgebra.
Its homogeneous components consist of differential forms along \(\varphi\). Mixed
forms are useful to represent characteristic classes and perform computations
of such.

An instance of this class is a mixed form along some differentiable map
\(\varphi: M \to N\) between two differentiable manifolds \(M\) and \(N\). More
precisely, a mixed form \(a\) along \(\varphi: M \to N\) can be considered as a
differentiable map

\[a: M \longrightarrow \bigoplus^n_{k=0} T^{(0,k)}N,\]

where \(T^{(0,k)}\) denotes the tensor bundle of type \((0,k)\), \(\bigoplus\)
the Whitney sum and \(n\) the dimension of \(N\), such that

More precisely, the exterior derivative on \(\Omega^k(M,\varphi)\) is a
linear map

\[\mathrm{d}_{k} : \Omega^k(M,\varphi) \to \Omega^{k+1}(M,\varphi),\]

where \(\Omega^k(M,\varphi)\) denotes the space of differential forms of
degree \(k\) along \(\varphi\)
(see exterior_derivative()
for further information). By linear extension, this induces a map on
\(\Omega^*(M,\varphi)\):

dest_map –
DiffMap
(default: None); destination map \(\Psi:\ U \rightarrow V\),
where \(V\) is an open subset of the manifold \(N\) where the mixed form
takes it values; if None, the restriction of \(\Phi\) to \(U\) is
used, \(\Phi\) being the differentiable map \(S \rightarrow M\) associated
with the mixed form

sage: M=Manifold(2,'M')# the 2-dimensional sphere S^2sage: U=M.open_subset('U')# complement of the North polesage: c_xy.<x,y>=U.chart()# stereographic coordinates from the North polesage: V=M.open_subset('V')# complement of the South polesage: c_uv.<u,v>=V.chart()# stereographic coordinates from the South polesage: M.declare_union(U,V)# S^2 is the union of U and Vsage: xy_to_uv=c_xy.transition_map(c_uv,(x/(x^2+y^2),y/(x^2+y^2)),....: intersection_name='W',restrictions1=x^2+y^2!=0,....: restrictions2=u^2+v^2!=0)sage: uv_to_xy=xy_to_uv.inverse()sage: W=U.intersection(V)sage: e_xy=c_xy.frame();e_uv=c_uv.frame()

rst – MixedForm of the same type as self, defined on
a subdomain of the domain of self

EXAMPLES:

Initialize the 2-sphere:

sage: M=Manifold(2,'M')# the 2-dimensional sphere S^2sage: U=M.open_subset('U')# complement of the North polesage: c_xy.<x,y>=U.chart()# stereographic coordinates from the North polesage: V=M.open_subset('V')# complement of the South polesage: c_uv.<u,v>=V.chart()# stereographic coordinates from the South polesage: M.declare_union(U,V)# S^2 is the union of U and Vsage: xy_to_uv=c_xy.transition_map(c_uv,(x/(x^2+y^2),y/(x^2+y^2)),....: intersection_name='W',restrictions1=x^2+y^2!=0,....: restrictions2=u^2+v^2!=0)sage: uv_to_xy=xy_to_uv.inverse()sage: W=U.intersection(V)sage: e_xy=c_xy.frame();e_uv=c_uv.frame()